Alina Dudeanu

Dimitar Jetchev

Joppe W. Bos

Abstract.
               We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm
problem in a general cyclic group 
                     
                        
                           𝐆
                        
                        $\mathbf {G}$
                     
                  . Unlike the setting studied by Kim et al.,
we consider additive walks: the setting used in practice to solve the elliptic curve discrete logarithm problem.
Our bounds differ from the birthday bound 
                     
                        
                           
                              𝒪
                              (
                              
                                 
                                    |
                                    𝐆
                                    |
                                 
                              
                              )
                           
                        
                        $\mathcal {O}(\sqrt{\vert \mathbf {G}\vert })$
                     
                  
by a factor of 
                     
                        
                           
                              
                                 log
                                 
                                    |
                                    𝐆
                                    |
                                 
                              
                           
                        
                        $\sqrt{\log {\vert \mathbf {G}\vert }}$
                     
                   and are based on mixing time estimates for random
walks on finite abelian groups due to Dou and Hildebrand.</jats:p

Collision bounds for the additive Pollard rho algorithm for solving discrete logarithms

Abstract

Similar works

Full text

Available Versions

Crossref